Since \(f\)  approaches a non-zero constant asymptotically, one requires that \(a\rightarrow n\)  asymptotically (which yields \((n-a)^2\,f^2 \rightarrow 0\) ) so that one avoids a logarithmic divergence in the energy of the vortex [1]}, [2]}. The Ricci scalar is a function of \(A\)  and \(B\)  and their derivatives:
\(R= \frac{ (B^{\prime })^2 A}{ 2 B^2}-\frac{ A^{\prime }}{r }-\frac{A^{\prime }B^{\prime }}{2 B}- \frac{B^{\prime } A}{r B} -\frac{ B^{\prime \prime } A}{B}\,.\) 
